AuthorTopic: 3D Geometry: The intersection of Math & Nature  (Read 1419 times)

Offline RE

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3D Geometry: The intersection of Math & Nature
« on: April 25, 2016, 10:27:20 PM »
Kicking off the thread with this vid on how to draw a dodecahedron with a compass and straight edge.  Quite remarkable.

This is one I never knew before.



<a href="http://www.youtube.com/v/gVdu5yU9MUk" target="_blank" class="new_win">http://www.youtube.com/v/gVdu5yU9MUk</a>

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Offline RE

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The intersection of Math & Nature: The Golden Rectangle
« Reply #1 on: May 01, 2016, 09:59:34 PM »









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Offline azozeo

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Re: 3D Geometry: The intersection of Math & Nature
« Reply #2 on: May 02, 2016, 12:52:52 PM »
Overlay that sequence of squares on the Mona Lisa starting with the eye.
That was Da Vinci's whole point he was trying to make with that painting.
Glad you posted this RE ....
I know exactly what you mean. Let me tell you why youíre here. Youíre here because you know something. What you know you canít explain, but you feel it. Youíve felt it your entire life, that thereís something wrong with the world.
You donít know what it is but its there, like a splinter in your mind

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Re: 3D Geometry: The intersection of Math & Nature
« Reply #3 on: May 02, 2016, 05:09:05 PM »
Overlay that sequence of squares on the Mona Lisa starting with the eye.
That was Da Vinci's whole point he was trying to make with that painting.
Glad you posted this RE ....

It's actually a sequence of Rectangles, not squares, and the Rectangle has to have dimensions in a very specific Ratio, the Golden Ratio.

Quote
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,

    \frac{a+b}{a} = \frac{a}{b} \ \stackrel{\text{def}}{=}\ \varphi,

where the Greek letter phi (\varphi or \phi) represents the golden ratio. Its value is:

    \varphi = \frac{1+\sqrt{5}}{2} = 1.6180339887\ldots. OEIS A001622

The golden ratio also is called the golden mean or golden section (Latin: sectio aurea).[1][2][3] Other names include extreme and mean ratio,[4] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[5] and golden number.[6][7][8]



A golden rectangle (in pink) with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship \frac{a+b}{a} = \frac{a}{b} \equiv \varphi.

The ratio is also demonstrated in the Pentagram, where the length of the sides of each successive pentagon have this relationship


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Offline azozeo

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Re: 3D Geometry: The intersection of Math & Nature
« Reply #4 on: May 02, 2016, 06:42:24 PM »
My mistake.
I've looked at a lot of Marty Leeds vids on this golden rule in nature.
I love this stuff. It's so seamless. The grand design is epic.
I know exactly what you mean. Let me tell you why youíre here. Youíre here because you know something. What you know you canít explain, but you feel it. Youíve felt it your entire life, that thereís something wrong with the world.
You donít know what it is but its there, like a splinter in your mind

 

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